Active optics enable modification of the wavefront shape for compensating aberrations due to, for example, thermal distortion of optical elements, optical part manufacturing and mounting errors, the gravity action or the absence of gravity in the space that distorts the results of an adjustment on the ground.
It is known a method for compensation of optical aberrations (Harvey J. E., Callahan G. M. Wavefront Error Compensation Capabilities of Multiactuator Deformable Mirrors//SPIE Vol. 141, Adaptive Optical Components, 1978, p. 50-57), providing for the measurement of wavefront aberrations and the transformation of the results of these measurements into a set of commands for actuators that deform the mirror.
For assigning a set of commands to force actuators, Zernike polynomials are the most often used (Active Optics Correction Of Thermal Distortion Of A 1.8 Meter Mirror/E. Pearson, L. Stepp, J. Fox//Optical Engineering, 27(2), 115-122 (Feb. 1, 1988). Axial displacements of the mirror reflective surface arising during mirror deformation are represented as the superposition of Zernike polynomials.
It is known a method for compensating optical aberrations disclosed in U.S. Pat. No. 8,102,583 providing formation of a set of commands for force actuators that deform the mirror, in order to move the reflective surface of this mirror in accordance with the superposition of the Zernike polynomials that compensate said optical aberrations.
A drawback of the above-mentioned method is a limited compensation accuracy of the wavefront aberrations. It is due to the fact that the Zernike polynomials describe displacements of optical surfaces, while strain state of a mirror is three-dimensional. That is why it is impossible to provide the superposition of the Zernike polynomials on the reflective surface of the deformable mirror to compensate the wavefront aberrations exactly.
At the same time, there exists some analogy between the Zernike polynomials used in the theory of optical aberrations and the orthogonal Clebsch polynomials that are used in the theory of elasticity (Lemaitre G.bR. Astronomical Optics and Elasticity Theory//Active Optics Methods, Berlin, Heidelberg: Springer, 2009, 587 p.). But the Clebsch polynomials are used only to describe displacements of thin round homogeneous plates. In the case of deformation of mirrors the geometric shape of which is different from the above described, the Clebsch polynomials, and consequently, the Zernike polynomials neither are suitable to describe the deformations of a mirror.
It is known as well a method for compensating optical aberrations disclosed in the paper by Cho M. K. Active optics performance study of the primary mirror of the Gemini Telescopes Project//Optical Telescopes of Today and Tomorrow, SPIE Vol. 2871, 1997, p. 272-290, in which the commands are assigned to force actuators in a way to provide, on the deformable mirror reflective surface, some displacements corresponding to the superposition of mirror natural frequency modes required for compensating the wavefront aberrations. Thanks to the consideration of the real geometric shape of the mirrors, this method can improve the compensation accuracy in comparison with the method using the Zernike polynomials, but only in part, since the natural frequency modes of the mirror vibrations used by the method do not correspond to the orthogonal modes of the mirror deformation in the conditions of quasi-static loading implemented in active optical systems. The last phenomenon is explained by the fact that the natural frequency modes of the mirror depend on the real distribution of the mass in the volume of the mirror structure, while in the case of the quasi-static loading, the inertia-mass characteristics have no effect on the displacements induced by force actuators.